Design of Sensorless Control System for Permanent Magnet Linear Synchronous Motor Based on Parametric Optimization Super-Twisting Sliding Mode Observer

Abstract

To improve the chattering problem caused by the terminal sliding mode control (TSMC) and sliding mode observer (SMO) in permanent magnet synchronous linear motor (PMLSM) control systems, this study presents the design of a continuous terminal sliding mode control (CTSMC) controller and an improved SMO. By enhancing the sliding mode surface, CTSMC enhances the dynamic response and robustness of the system. The observer replaces the traditional sliding mode switching law with a super-twisting (ST) algorithm, a twisting algorithm that makes use of the structural characteristics of the second-order sliding mode to ensure output continuity and reduce the observed buffeting. The sliding mode gain of the ST algorithm is optimized using the particle swarm optimization (PSO) algorithm to acquire the optimal parameters and fully exploit the observer’s performance potential. Finally, the proposed method is simulated and tested. The comparison results show that the proposed method boosts the system’s dynamic response and robustness and reduces chattering.

1. Introduction

With the continuous progress of industry, the requirements of motors continue to increase. Due to its simple structure, high positioning accuracy, large thrust strength, and lack of a transmission system, PMLSM is attracting more and more attention and research [1,2]. However, achieving high precision control of PMLSM is challenging due to the serious coupling between states, the uncertainty of load disturbance, and the resultant side effects that can affect its control performance [3,4]. To address the control problem of PMLSM, several new control algorithms, including model-free control, active disturbance rejection control, model predictive control, and sliding mode control (SMC), have been advocated [5,6]. Of these, SMC control is essentially a nonlinear control method, which is not affected by external disturbances and parameter changes and has extremely high robustness and extremely fast response speed. With the change in system state, the SMC structure also changes, so that the system will slide in accordance with the expected state trajectory and gradually converge to the sliding mode surface and ultimately stabilize [7,8]. However, traditional SMC cannot guarantee that the system converges within a finite time frame, and because the switching frequency of SMC cannot reach infinite speed, serious chattering problems occur [9,10]. In the literature [11], a TSMC is designed to solve the problem that the traditional SMC cannot guarantee that the system converges within a finite time frame, to reduce the convergence time, and to improve the dynamic performance, but the system produces strange phenomena in a certain area. In the literature [12], a neural network is combined with SMC, and the switching gain in the SMC is trained by the neural network, which effectively improves the stability and robustness of the system and reduces chattering. However, the high cost of the neural network is not conducive to practical applications. In the literature [13], an adaptive SMC is designed that can effectively suppress chattering and reduce tracking errors by dynamically adjusting the width and gain of the boundary layer functions. However, the adaptive control increases the calculation and adjustment time. In the literature [14], a Gaussian error function is used instead of a sign function to reduce chattering. However, this method has high requirements for the precision of various state quantities of the motor, which is not conducive to practical applications.

In most servo control systems, precise actuator position and speed feedback is essential for achieving high-performance control [15]. Traditional PMLSM relies on sensors to obtain position and speed information, but this brings a high cost and limited working place, and the control performance is limited by detection accuracy and other problems [16,17]. To decrease the cost of the system and expand the working place of the motor, sensorless control strategies have been widely used in PMLSM. Sensorless control mainly comprises an extended Kalman filter algorithm, a model reference adaptive algorithm, an SMO algorithm, and an artificial intelligence theoretical estimation algorithm [6,18,19]. Among them, SMO is famous for its easy convergence, insensitivity to motor parameters, simple and convenient parameter adjustment, less computation, etc., and has been widely used in practice [20,21,22]. The literature [23] designed an adaptive sliding mode observer that can adjust sliding mode gain adaptively according to the system state and improve the responsiveness and robustness. However, it does not solve the inherent chattering problem of SMO, which seriously affects the velocity and location estimation accuracy in the motor. In the literature [24], a sinusoidal saturation function is utilized to substitute the sign function, which reduces system chattering caused by the sliding mode observer to a certain extent, but also undermines the system’s stability and performance. In the literature [25], a high-order integrated sliding mode surface is employed to suppress chattering, which achieves good results. However, this method has the defects of a complex algorithm and too many debugging parameters.

To address the singularity issue of TSMC and improve responsiveness and robustness, a CTSMC was designed. The use of the CTSMC guarantees that the system state will converge to the desired trajectory within a bounded time frame, avoids the occurrence of singularity, improves the ability of the system to deal with disturbance, and ensures the continuity of control. To address the chattering issue of traditional SMOs, an ST-SMO with optimized parameters was designed. The ST algorithm is utilized to substitute the conventional sliding mode switching law, and the second-order sliding mode structural characteristics in the ST are used to ensure output continuity and reduce chattering. The PSO is used to optimize the sliding mode gain of the ST algorithm, and a set of optimal parameters is obtained to achieve better control performance. The simulation and experimental results demonstrate the practicability of the proposed strategy. The key innovation points of this article are as follows:

(1)

Simplify the control structure. The system’s control structure is simplified by integrating the position loop and speed loop, and the position controller is developed using the CTSMC algorithm.

(2)

Improve tracking accuracy. The ST algorithm is designed and the PSO is used to optimize the sliding mode gain of the ST algorithm, and a set of optimal parameters is obtained to better realize the control performance of the algorithm.

(3)

System comparison and validation. By comparing the designed CTSMC with the SMC and TSMC systems, and by comparing the designed PSO super-twisting sliding mode control (PSO-ST-SMC) with the SMO, the advantage of the designed system is verified.

2. Mathematical Model of PMLSM

In terms of structure, the PMLSM can be regarded as a rotating motor, as shown in Figure 1, which is split and spread along the radial direction. In PMLSM, the movable structure is called the moving part and the fixed part is called the stator.

To simplify analysis and control system design, the following assumptions are usually made during the construction of the mathematical model of a PMLSM: motor core saturation is ignored, inductance parameters remain unchanged, eddy current and hysteresis losses in the motor are ignored, the rotor magnetic field is sinusoidal in the air gap space, there is no damping winding on the rotor, and the effect of external rings on motor performance is not considered. Based on the above assumptions, the mathematical model of PMLSM under the d-q axis was established:

$$\left\{\begin{array}{c}{u}_d=R{i}_d+L_d\frac{d{i}_d}{dt}-\frac{\pi }{\tau }v{L}_q{i}_q\\ u_q=R{i}_q+L_q\frac{d{i}_q}{dt}+\frac{\pi }{\tau }v(L_d{i}_d+{\psi }_f)\end{array}\right.,$$

(1)

where $R$ is the stator resistance, $\tau$ is PMLSM pole distance, $\nu$ is the speed of PMLSM, $u_d$ and $u_q$ are the voltage of PMLSM, $i_d$ and $i_q$ are the current of PMLSM, $L_d$ and $L_q$ are the inductance components of PMLSM, and ${\psi }_f$ is the permanent magnet flux link.

The electromagnetic thrust equation for PMLSM is

$$F_e=p_n\frac{3\pi }{2\tau }[{\psi }_f{i}_q+(L_d-L_q\left)i_d{i}_q\right],$$

(2)

where $F_e$ is the PMLSM thrust and $p_n$ is the polar number of PMLSM.

In PMLSM, $L_d=L_q$, and the equation can be streamlined to

$$F_e=p_n\frac{3\pi }{2\tau }{\psi }_f{i}_q=K_f{i}_q,$$

(3)

where $K_f$ is the electromagnetic thrust coefficient and $K_f=p_n\frac{3\pi }{2\tau }{\psi }_f$.

The kinetic equation of PMLSM is

$$F_e=M\frac{dv}{dt}+Bv+F_d$$

(4)

where $M$ is the moving mass, $B$ is the viscous friction factor, $v$ is the motion velocity of PMLSM, and $F_d$ is the sum of system uncertainties.

According to Equation (4), the equation of motion of PMLSM system is:

$$\frac{dv}{dt}=\frac{K_f{i}_q}{M}-\frac{Bv}{M}-\frac{F_d}{M}=au+bv+c{F}_d,$$

(5)

where $a=\frac{K_f}{M}$, $b=-\frac{B}{M}$, $c=-\frac{1}{M}$, and $u=i_q$.

3. Position Controller Design

3.1. Design of the CTSMC Position Controller

To address the issue of PMLSM location tracking control, a CTSMC controller was designed, which can guarantee good robustness and tracking performance of the closed-loop system.

The PMLSM’s CTSMC controller was designed based on position error feedback. The system input is the position error and the output is the current reference value. Define the system state variable as

$$\left\{\begin{array}{c}{e}_1=x_r-x\\ e_2={\dot{e}}_1={\dot{x}}_r-v\\ {\dot{e}}_2={\ddot{x}}_r-\dot{v}\end{array}\right.,$$

(6)

where $x_r$ is the reference position value, smooth everywhere and differentiable in the second order, and $x$ is the actual position. By substituting Equation (5) into Equation (6), we obtain:

$${\dot{e}}_2={\ddot{x}}_{ref}-\dot{v}={\ddot{x}}_r-au-bv-c{F}_d.$$

(7)

To enhance the system’s robustness and dynamic performance, the CTSMC sliding mode surface is formulated as follows:

$$s=e_1+{\beta }_1{\left|e_2\right|}^{y_1}sgn\left(e_2\right),$$

(8)

where ${\beta }_1>0, 1<y_1<2$, these two coefficients are constants to be designed. The derivation of Equation (8) can be obtained as follows:

$$\dot{s}=e_2+{\beta }_1{y}_1{\left|e_2\right|}^{y_1-1}{\dot{e}}_2.$$

(9)

This paper adopts an equivalent control method. The structure of the equivalent control law is:

$$u=u_{eq}+u_{sw},$$

(10)

where $u_{eq}$ is the slipping-mode equivalent control part and $u_{sw}$ represents the uncertain component of the control system.

The motion process in the control system is divided into two stages, the approach stage and the sliding mode. When the system is in the sliding mode, the equivalent control part plays a role, and the system is completely located in the sliding mode region on the sliding mode surface. The essential condition that the state trajectory stays in the sliding mode surface is $\dot{s}=0$, making $\dot{s}=0$ and $F_d=0$ equivalent control parts:

$$\begin{array}{cc}\dot{s}& =e_2+{\beta }_1{y}_1{\left|e_2\right|}^{y_1-1}{\dot{e}}_2\hfill \\ & =e_2+{\beta }_1{y}_1{\left|e_2\right|}^{y_1-1}({\ddot{x}}_{ref}-a{i}_q-bv-c{F}_d)\hfill \\ & =0\hfill \end{array},$$

(11)

$$u_{eq}=\frac{1}{a}({\ddot{x}}_r-bv+\frac{1}{{\beta }_1{y}_1}{\left|e_2\right|}^{2-y_1}sgn(e_2\left)\right).$$

(12)

The switching control $u_{sw}$ plays a role as the system reaches the sliding mode; $u_{sw}$ makes the system state move toward the sliding mode surface by controlling high-frequency switching and ensures that the state will slide along the sliding mode surface to the stable state. Switching control $u_{sw}$ can be designed as:

$$u_{sw}=\frac{1}{a}\left(\epsilon \mathit{sgn}(s\right)+ks),$$

(13)

where the epsilon $\epsilon \mathit{sgn}(s)$ is uniform convergence and $ks$ is exponential convergence. The CTSMC position controller is obtained as follows:

$$u=u_{eq}+u_{sw}=\frac{1}{a}({\ddot{x}}_r-bv+\frac{1}{{\beta }_1{y}_1}{\left|e_2\right|}^{2-y_1}sgn(e_2)+\epsilon \mathit{sgn}(s)+ks).$$

(14)

3.2. Proof of Stability of CTSMC

In order to ensure that the system state motion trajectory is asymptotically stable after reaching the sliding mode region, it will strictly move along the predetermined sliding surface to the steady equipoise point, which is verified with the Lyapunov function. The Lyapunov function is defined as follows:

$$V=\frac{1}{2}{s}^2.$$

(15)

Taking the derivative of the Lyapunov function yields:

$$\dot{V}=s\dot{s}.$$

(16)

By substituting Equations (7), (9), and (14) into Equation (16), we obtain:

$$\begin{array}{cc}\dot{V}=s\dot{s}=& s(-{\beta }_1{y}_1{\left|e_2\right|}^{y_1-1}(\epsilon \mathit{sgn}(s)+ks))\hfill \\ & \le -{\beta }_1{y}_1{\left|e_2\right|}^{y_1-1}(\left|s\right|+k{s}^2)\hfill \end{array}.$$

(17)

Because the ${\beta }_1>0,$ $1<y_1<2$, it can be learned that ${\left|e_2\right|}^{y_1-1}>0$, available $\dot{V}\le 0$. Based on the Lyapunov stability theory, the system state asymptotically approaches $s=0$. Therefore, the displacement feedback $x$ of the motor will converge to the desired displacement signal $x_{ref}$ in a finite time. The control structure based on the CTSMC method is shown in Figure 2.

4. Observer Design

4.1. Traditional SMO

Traditional SMOs are designed in the static coordinate system $\alpha -\beta$; the current state equation is:

$$\left\{\begin{array}{c}\frac{d{i}_{\alpha }}{dt}=\frac{1}{L}(-R{i}_{\alpha }+u_{\alpha }-E_{\alpha })\\ \frac{d{i}_{\beta }}{dt}=\frac{1}{L}(-R{i}_{\beta }+u_{\beta }-E_{\beta })\end{array}\right.,$$

(18)

where $E_a=-\frac{\pi }{\tau }v{\psi }_f E_{\beta }=-\frac{\pi }{\tau }v{\psi }_f \cos \theta$ is the induced electromotive force in coordinate system $\alpha -\beta$. To acquire an estimate of the extended back electromotive force, a conventional SMO can be designed as

$$\left\{\begin{array}{c}\frac{d{\widehat{i}}_{\alpha }}{dt}=\frac{1}{L}(-R{\widehat{i}}_{\alpha }+u_{\alpha }-z_{\alpha })\\ \frac{d{\widehat{i}}_{\beta }}{dt}=\frac{1}{L}(-R{\widehat{i}}_{\beta }+u_{\beta }-z_{\beta })\end{array}\right.$$

(19)

$$\left\{\begin{array}{c}{z}_{\alpha }=k\mathit{sgn}({\widehat{i}}_{\alpha }-i_{\alpha })\\ z_{\beta }=k\mathit{sgn}({\widehat{i}}_{\beta }-i_{\beta })\end{array}\right.,$$

(20)

where ${\widehat{i}}_{\alpha }$, ${\widehat{i}}_{\beta }$ are the current observations and $k$ is the gain.

Subtract from Equation (18), and the equation for the current error state is:

$$\left\{\begin{array}{c}\frac{d{\tilde{i}}_{\alpha }}{dt}=\frac{1}{L}(-R{\tilde{i}}_{\alpha }-z_{\alpha }+E_{\alpha })\\ \frac{d{\tilde{i}}_{\beta }}{dt}=\frac{1}{L}(-R{\tilde{i}}_{\beta }-z_{\beta }+E_{\beta })\end{array}\right.,$$

(21)

where ${\tilde{i}}_{\alpha }={\widehat{i}}_{\alpha }-i_{\alpha }$ and ${\tilde{i}}_{\beta }={\widehat{i}}_{\beta }-i_{\beta }$ are the current observation errors.

When a state variable is located on a sliding mode,

$$\tilde{i}=\dot{\tilde{i}}=0.$$

(22)

By substituting Equation (22) into Equation (21), we obtain

$$E=[k\mathit{sgn}({\widehat{i}}_a-i_a)k\mathit{sgn}({\widehat{i}}_{\beta }-i_{\beta }){]}^T.$$

(23)

It can be seen from Equation (23) that there is a high-frequency switching signal in the estimated back electromotive force. Nevertheless, there is a discontinuous high-frequency switching signal in the estimated back electromotive force, which will produce a large deviation when the position velocity information is obtained from the back electromotive force.

4.2. ST Algorithm

In the traditional SMO, due to the existence of the sign function, a serious chattering phenomenon is caused, which leads to a large tracking error. While low-pass filtering can reduce chattering, it can also introduce phase delay. Therefore, the ST algorithm is presented to address the chattering issue in the traditional SMO. The ST form is illustrated in Equation (24):

$$\left\{\begin{array}{c}\frac{d{x}_1}{dt}=k_1{\left|{\tilde{x}}_1\right|}^{\frac{1}{2}}\mathit{sgn}({\tilde{x}}_1)+x_2\hfill \\ \frac{d{x}_2}{dt}=k_2\mathit{sgn}({\tilde{x}}_1)\hfill \end{array}\right.,$$

(24)

where $x_1,x_2$ are state variables, ${\tilde{x}}_i={\widehat{x}}_i-x_i$ is the error between the estimated value and the actual value, and $k_1,k_2$ are gains. The $x_1$ in the superhelix algorithm is substituted by the current signal ${\widehat{i}}_{\mathsf{\alpha }},{\widehat{i}}_{\beta }$ estimated by PMLSM, respectively, to obtain the PMLSM sliding mode observer based on the superhelix algorithm, as illustrated in Equation (25):

$$\left\{\begin{array}{c}\frac{d{\widehat{i}}_{\alpha }}{dt}=\frac{1}{L}(-R{\widehat{i}}_{\alpha }+u_a-\int k_{1}sgn\left({\tilde{i}}_{\alpha }\right)dt-k_2{\left|{\tilde{i}}_{\alpha }\right|}^{\frac{1}{2}}sgn({\tilde{i}}_{\alpha }\left)\right)\\ \frac{d{\widehat{i}}_{\beta }}{dt}=\frac{1}{L}(-R{\widehat{i}}_{\beta }+u_{\beta }-\int k_{1}sgn\left({\tilde{i}}_{\beta }\right)dt-k_2{\left|{\tilde{i}}_{\beta }\right|}^{\frac{1}{2}}sgn({\tilde{i}}_{\beta }\left)\right)\end{array}\right..$$

(25)

The error equation of the current obtained by subtracting Equation (18) from Equation (25) is:

$$\left\{\begin{array}{c}\frac{d{\tilde{i}}_{\alpha }}{dt}=\frac{1}{L}(-R{\tilde{i}}_{\alpha }-\int k_{1}sgn\left({\tilde{i}}_{\alpha }\right)dt-k_2{\left|{\tilde{i}}_{\alpha }\right|}^{\frac{1}{2}}sgn({\tilde{i}}_{\alpha })+E_{\alpha })\\ \frac{d{\tilde{i}}_{\beta }}{dt}=\frac{1}{L}(-R{\tilde{i}}_{\beta }-\int k_{1}sgn\left({\tilde{i}}_{\beta }\right)dt-k_2{\left|{\tilde{i}}_{\beta }\right|}^{\frac{1}{2}}sgn({\tilde{i}}_{\beta })+E_{\beta })\end{array}\right.,$$

(26)

where ${\tilde{i}}_{\alpha }={\widehat{i}}_{\alpha }-i_{\alpha }$ and ${\tilde{i}}_{\beta }={\widehat{i}}_{\beta }-i_{\beta }$ are current observation errors.

When the system is steady, the estimation is equivalent to the actual current, namely, ${\tilde{i}}_{\alpha }=0$, ${\tilde{i}}_{\beta }=0$. At this time, the back electromotive force of PMLSM can be obtained:

$$\left\{\begin{array}{c}{E}_a=\int k_{1}sgn\left({\tilde{i}}_{\alpha }\right)dt+k_2{\left|{\tilde{i}}_{\alpha }\right|}^{\frac{1}{2}}sgn\left({\tilde{i}}_{\alpha }\right)\\ E_{\beta }=\int k_{1}sgn\left({\tilde{i}}_{\beta }\right)dt+k_2{\left|{\tilde{i}}_{\beta }\right|}^{\frac{1}{2}}sgn\left({\tilde{i}}_{\beta }\right)\end{array}\right.,$$

(27)

where $k_1,k_2$ are sliding mode gains.

The ST algorithm is a second-order sliding mode and can hide the discontinuous switching in the integral to ensure that the control law is continuous in time, greatly reducing chattering. The ST algorithm still has a high-frequency switching function to ensure the robustness compared with other high-order sliding modes; it is easy to implement and has high practical value.

However, the sliding mode gain $k_1$, $k_2$ in the ST algorithm is difficult to obtain; it is usually obtained manually through repeated debugging, and the requirements for parameters are different when facing different input waveforms. If one parameter is debugged for each input waveform, it will not only consume energy but also not be conducive to practical applications. Moreover, we have no way to know whether the manually debugged parameters are exerting the optimal performance of the controller. In order to save the cost and energy and exert the optimal performance of the controller, the PSO algorithm is introduced to find the optimal parameter value. The diagram is shown in Figure 2.

4.3. PSO Algorithm

PSO is an optimization algorithm based on swarm intelligence. It was proposed by American sociologists Kennedy and Eberhart to observe the behavior of animal groups with swarm behavior, such as birds or fish. It has the advantages of fast convergence speed and few adjustment parameters, and is simple and easy to implement. The core idea of the PSO algorithm is to simulate the behavior of animal groups with clustering behavior, such as birds or fish schools. Through information exchange and cooperation between individuals, the global optimal solution is sought. Each possible solution in the parameter space to be optimized is regarded as a particle, and the optimal solution is sought by constantly updating the position and speed of each particle. Its specific implementation process is shown in the figure below:

(1)

The PSO algorithm begins by initializing the particle swarm, which includes the population size, initial position and velocity of particles, and other relevant parameters.

(2)

It evaluates the fitness value of each particle based on its current position and velocity.

(3)

The fitness value of the particle is compared with the current best personal value, and the better value is chosen as the best personal value.

(4)

The best personal value of each particle is compared with the global extreme value, and the superior one is chosen as the global extreme value.

(5)

The particle position and velocity update according to the formula.

(6)

The PSO algorithm judges whether it meets the maximum number of iterations or the set termination condition. If either of these conditions is met, it exits the optimization process and outputs the optimal solution; if the conditions are not met, it returns to Step 2.

In the PSO algorithm, the updating of particle velocity and position is achieved with the following formula:

$$\left\{\begin{array}{c}{v}_i(t+1)=w{v}_i\left(t\right)+c_1{r}_1(pbs{t}_i-x_i(t\left)\right)+c_2{r}_2(gbest-x_i(t\left)\right)\hfill \\ x_i(t+1)=x_i\left(t\right)+v_i(t+1)\hfill \end{array}\right.,$$

(28)

where $v_i\left(t\right)$ represents the speed of particle $i$ at time $t$, $x_i\left(t\right)$ represents the position of particle $i$ at time $t$, $pbs{t}_i$ represents the personal best value of particle $i$, $gbest$ represents the global best value, $w$ is the inertia weight, $c_1$ and $c_2$ are two constants, $c_1$ is the self-learning factor, $c_2$ is the group learning factor, and $r_1$ and $r_2$ are random numbers between 0 and 1.

Firstly, the PSO generates a particle swarm, and the particles in the particle swarm are assigned to the parameters $k_1$, $k_2$ in the controller. Then, the control system Simulink model is run to constantly update the position and speed of the particle swarm and output the corresponding performance indicators of this set of parameters. When the largest number of iterations or the required performance indicators are reached, it returns the values of parameters $k_1$, $k_2$.

In this paper, ITAE is selected as the error evaluation index to evaluate the performance index of each group of parameters. The error evaluation index is defined as:

$$J={\int }_0^{\infty }t\left|e_1\right|dt,$$

(29)

where $e_1$ represents the difference between the estimated current and the actual current of the PMLSM. The diagram is shown in Figure 3.

By combining the PSO algorithm with the ST algorithm, in which the PSO algorithm is used to optimize the sliding mode gain $k_1$, $k_2$ of the ST algorithm, find the optimal parameter values, and rederive the formula, the back electromotive force of PMLSM can be written as follows:

$$\left\{\begin{array}{c}{E}_a=\int k_{1s}sgn\left({\tilde{i}}_{\alpha }\right)dt+k_{2s}{\left|{\tilde{i}}_{\alpha }\right|}^{\frac{1}{2}}sgn\left({\tilde{i}}_{\alpha }\right)\\ E_{\beta }=\int k_{1s}sgn\left({\tilde{i}}_{\beta }\right)dt+k_{2s}{\left|{\tilde{i}}_{\beta }\right|}^{\frac{1}{2}}sgn\left({\tilde{i}}_{\beta }\right)\end{array}\right..$$

(30)

In the formula, $k_{1s}$ and $k_{2s}$ are parameters obtained by the particle swarm optimization algorithm.

A low-pass filter is typically added to obtain a continuous estimation of the back electromotive force, as shown in Equation (31):

$${\widehat{E}}_{\alpha \beta }=\frac{{\omega }_c}{s+{\omega }_c}{E}_{\alpha \beta }$$

(31)

where ${\omega }_c=2\mathsf{\pi }{f}_c$, and $f_c$ represents the cutoff frequency of the low-pass filter.

With the estimated back electromotive force in hand, the location information of the actuator can be obtained using the following formula. The estimated value of the location can be expressed as:

$$\widehat{s}=\int \frac{\tau \sqrt{{{\widehat{E}}_{\alpha }}^2+{{\widehat{E}}_{\beta }}^2}}{\pi {\psi }_f}dt.$$

(32)

In summary, the control principle of the designed observer is as follows: the optimal gain of the observer is first obtained by the PSO algorithm, then the estimated back electromotive force signal is obtained by the ST algorithm, and finally the position signal is obtained by the phase-locked loop module.

5. Simulation and Experiment

In this paper, a sensorless control system based on PSO-ST-SMO is proposed, and its block diagram is illustrated in Figure 4.

Table 1. PMLSM main parameters.

Parameter	Numerical Value
Stator resistance, Rs/Ω	2.6
d-q axis inductance, Ldq/mH	6.27
Moving mass, m/kg	1.425
Coefficient of viscous friction, B/N/m⋅s	0.2
Pole distance, τ/m	0.018
Flux link, wb	0.24
DC Bus Voltage, U/V	48

shows the PMLSM arguments. To emphasize the superiority of the system, a comparative experiment was set up, and the SMC, TSMC, and CTSMC were set up, respectively, to compare the controllers. On the basis of the CTSMC controller, the traditional SMO and PSO-ST-SMO are integrated for observer comparison.

The PSO parameters are as follows: iteration number Maxiter = 50, population number swarm size = 30, particle dimension (number of parameters to be adjusted) Dim = 2, acceleration coefficient c1 = c2 = 2, and the inertia coefficient w = 0.8. The cutoff condition of the algorithm is whether the largest number of iterations is reached or whether the error is less than the set value after the position is stabilized. After several iterations, the optimal parameters of the ST-NTSMC position controller obtained by the PSO algorithm and the corresponding minimum value of fitness function are shown in

Table 2. Parameter optimization result.

Algorithm	k1	k2	ƒ (Swarm)
PSO	400.59	248.42	0.571

. The process of parameter convergence is shown in Figure 5a, and the process of fitness function convergence is shown in Figure 5b.

5.1. Comparison between Position Controllers

To validate the tracking efficiency of the control system, no-load and load experiments were carried out, and three sets of simulation conditions were set:

(1)

The change in the given reference position is 0.1 m–0.3 m.

(2)

Given a reference position of 0.2 m, load the system abruptly.

(3)

A sinusoidal wave with a fixed amplitude of 0.2 m is applied to the system.

(1)

The change in the given reference position is 0.1 m–0.3 m.

The PMLSM was started under a no-load condition, the expected starting position of the trajectory was 0.1 m, and the position changed by 0.1 m every 2 s. The performance of the different control methods was compared.

The position tracking curve of PMLSM under the control of SMC, TSMC, and CTSMC is shown in Figure 6. PMLSM tracks the expected signal under the no-load condition, the system runs for 10 s, the starting position is 0.1 m, and the position changes by 0.1 m every 2 s. Upon closer inspection of Figure 6a, it becomes apparent that the three control schemes demonstrate effective tracking of the desired position. In the ascent stage, the response speed of CTSMC is obviously faster than the other two controls, the system adjustment time is shortest, the given signal can be quickly tracked, and the dynamic response capability is the best. Figure 6 presents the position tracking error curves for all three control schemes. As can be seen from the local magnification of Figure 6b, SMC has the worst tracking performance and causes the maximum system chattering when tracking the desired position. In comparison, TSMC and CTSMC have better tracking performance and better control accuracy. When the position changes, CTSMC and TSMC adjust for about 0.2 s and 0.3 s, reflecting better dynamic performance of the control system based on the CTSMC controller. In summary, CTSMC has the best tracking effect on the desired trajectory and can track the desired trajectory quickly when it changes, with minimum buffeting and the highest tracking and positioning accuracy.

(2)

Given a reference position of 0.2 m, load the system abruptly.

To validate the robustness of the control system, a 45 N load was randomly added to the system.

During the operation of the system, a 45 N load was randomly added. Figure 7a shows the operation of the system. As can be observed from the local magnification in Figure 7a, when compared with SMC and TSMC, CTSMC’s position changes are significantly reduced, and the adjustment time is faster. Figure 7b shows the error comparison of the three control schemes under load. Through local amplification of Figure 7b, it can be seen that after the 45 N load is added abruptly, the maximum position error for SMC is approximately 0.012 m and the maximum position errors of the TSMC and CTSMC are approximately 0.005 m and 0.001 m, respectively. Figure 7c shows the comparison of the electromagnetic thrust under load. According to the local amplification of Figure 7c, the CTSMC’s electromagnetic thrust is almost 0 under no-load conditions, and the system runs stably. After the sudden addition of the 45 N load, the electromagnetic thrust fluctuation of CTSMC is small and the control is more stable. To sum up, CTSMC has stronger robustness. After the system is suddenly loaded, the error fluctuation is small and the recovery time is fast, so that the motor can run smoothly. In conclusion, CTSMC significantly improves the anti-disturbance performance.

(3)

A sinusoidal wave with a fixed amplitude of 0.2 m was given to add load to the system.

A sinusoidal curve with an expected trajectory of $s=0.2\ast \mathit{sin}t$ adds 45 N load to the system at 4.6 s.

At 4.6 s, a 45 N load disturbance was abruptly added to the system to test the robustness of the system. Figure 8a displays the position tracking curves of the system under all three control schemes. All three control schemes are able to effectively track the given position curve even under the presence of the load disturbance, demonstrating a certain degree of robustness. Nevertheless, as apparent from the local magnification in Figure 8a, there is a large fluctuation in the SMC control tracking curve at 4.6 s. Meanwhile, the tracking curves of TSMC and CTSMC fluctuate less. Figure 8b is the position error curve. The local amplification of Figure 8b shows that the maximum errors of SMC, TSMC, and CTSMC are about 0.006 m, 0.004 m, and 0.001 m, respectively. The position error fluctuation of the CTSMC scheme is the least, and exhibits superior tracking accuracy and stronger robustness when compared with the other two control schemes. In conclusion, the CTSMC scheme shows strong system robustness in the presence of interference.

5.2. Comparison of Observers

The control performance of the CTSMC system proposed in this paper was verified by a controller simulation comparison. Building on the CTSMC system, traditional SMO was incorporated and compared with the proposed PSO-ST-SMO.

From Figure 9a–c, it is evident that traditional SMO results in significant chattering in the estimation signal. The observer transmits the error caused by chattering to the position controller of the control system, thus reducing the overall accuracy of the control system. Figure 9d–f shows the designed P-ST-SMO system. The ST algorithm is designed to substitute the conventional sliding mode control law, and the PSO is introduced at the same time to obtain the optimal parameters, give full play to the performance of the observer, effectively decrease the observation chattering, and enhance the tracking accuracy of the system.

5.3. Experiment

To further validate the feasibility and effectiveness of the control system, the block diagram of the PMLSM control system based on STM32 is illustrated in Figure 10 as built.

The rectifier and intelligent circuit form the power module (IPM) to supply power to the entire system. Real-time monitoring and processing of the system are achieved by utilizing detection circuits such as a current sensor and grating sensor. Figure 11 shows the construction of the experimental platform.

Firstly, the CTSMC control system proposed in this paper was established, and the traditional SMC and TSMC control systems were compared to analyze the lively performance and anti-disturbance capacity of the designed controller. The experimental conditions are consistent with the simulation. Through the on/off signal control of the electromagnet, the loading action of the weight load can be quickly completed.

(1)

Controller Comparison (No-Load)

Figure 12a–c indicates the comparison of the system running position and trajectory, and Figure 12d–f shows the performance indicators when tracking the expected trajectory, which are, respectively, overshoot, maximum error, and average error. Black is the expected trajectory and red is the tracking trajectory of the system. Local magnification of Figure 12a–c shows that the proposed CTSMC has the highest control accuracy and the minimum system buffeting. It can be seen from Figure 12d–f that the proposed CTSMC performance index is optimal. The specific data are shown in

Table 3. Control method performance indicators (Step).

Error Type	SMC	TSMC	CTSMC
Expected trajectory	Step	Step	Step
Overshoot (m)	0.0035	0.0021	0.0009
Maximum error (m)	0.0042	0.0031	0.0015
Average error (m)	0.0037	0.0029	0.0012

. It can be seen from the data that the designed control method is better than the other two methods in terms of overshoot, maximum error, and average error.

(2)

Controller Comparison (Load)

Figure 13a–c is the comparison of the position trajectory of the system when the step signal is tracked; Figure 13d–f is the error fluctuation of three control schemes when the step signal is tracked; Figure 13g–i is the comparison of the position trajectory of the system when the sinusoidal signal is tracked; and Figure 13j–l is the error fluctuation of the three control schemes when the sinusoidal signal is tracked. The waveform change sizes of the SMC, TSMC, and CTSMC after a sudden load are about 0.015 m, 0.008 m, and 0.002 m, respectively, and the recovery times are about 0.4 s, 0.25 s, and 0.1 s, respectively.

As can be seen from Figure 13, the traditional SMC not only has a long adjustment time but also poor tracking accuracy in the acceleration process without load. Moreover, when loading is added, the position waveform changes more obviously. Compared with the traditional SMC control system, the adjustment time of the TSMC control system is shorter, the overshoot is smaller, and the position waveform changes only slightly after adding the load. Finally, the position waveform of the CTSMC control system developed in this paper is analyzed. In the position rise stage, the system has almost no overshoot, the rise time and adjustment time are shorter, the position waveform jitter is small, and the control precision is high. After loading is added, the location is barely affected by the load.

(3)

Observer Comparison

The first part of the experiment was to compare and verify the installation of the encoder, which proves the good control performance of the CTSMC position controller. The second half of the experiment was based on the designed CTSMC control system, adding the traditional SMO and PSO-ST-SMO for comparison. The feasibility and effectiveness of the designed PSO-ST-SMO were analyzed. Given three waveforms for the reference position, the position observed by the observer is returned to the controller, and the position waveform is obtained, as shown in Figure 14.

Figure 14a–c shows the position trajectory comparison of SMO, and Figure 14d–f shows the position trajectory comparison of PSO-ST-SMO. Because the SMO is accompanied by high-frequency chattering in sliding mode, the estimated back electromotive force also has chattering. The traditional SMO system introduces chattering directly into the arctangent function division operation, leading to an amplification of the error resulting from the chattering. In the PSO-ST-SMO control system developed in this paper, the ST algorithm is designed to substitute the conventional sliding mode control law, and PSO is introduced to obtain the optimal parameters, give full play to the performance of the observer, and effectively weaken chattering.

6. Conclusions

To enhance the position tracking precision of PMLSM, a CTSMC position controller is designed and a PSO-ST-SMO is used to replace the conventional SMO, which decreases the chattering of the system. Through the comparison of the simulation and experiment, the CTSMC speeds up the lively response of the system, reduces the position overshoot, strengthens the robustness of the system, and realizes the fast tracking of the position. PSO-ST-SMO realizes position observation, reduces system chattering, and enhances system observation precision. Both the simulation and experimental findings demonstrate that the control precision is higher, the observation error is smaller, the control effect is better, and the control performance of the PMLSM is generally improved.